Timeline for Quantiles in mixture distributions - mathematical explanation

Current License: CC BY-SA 3.0

7 events

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Sep 14, 2017 at 22:36	comment	added	Glen_b		ctd... Perhaps bounded by some small positive value would do, but for it to be practically useful (and again I'm only considering the right tail but similar things flipped around would apply on the left) you'd probably want that ratio going to zero pretty quickly. I wonder if something like $\lim_{x\to\infty} P(X>x+t)/P(X>x)\leq ke^{-ct}$ for some problem-dependent $k$ and $c$ would be enough for a practical definition? I guess I'd call that "asymptotically exponential tails"; I'm not sure that would be quite be light enough to be really practically useful, though it's perhaps in the area
Sep 14, 2017 at 22:16	comment	added	Glen_b		Actually, even asymmetric would do if both tails weren't heavy. How thin the tails need to be really depends but I guess that asymptotically you'd be looking for (loosely) something like $\lim_{x\to\infty} P(X>x+t)/P(X>x) \to 0$ (but I've left that too vague for a formal definition). From the Wikipedia article on Heavy tailed distributions we'd be looking for something not long-tailed, because then as you went far out the proportion would go to 1 instead of 0. ... ctd
Sep 14, 2017 at 21:03	vote	accept	Alexander Whyte
Sep 14, 2017 at 21:00	comment	added	Alexander Whyte		This is good; I now understand the mechanics of what is going on. Do you think it is possible to generalise the result in some way? I am thinking that this will hold as long as the original distribution is symmetric and has thin tails. Is there some technical term that can be applied to such distributions?
Sep 14, 2017 at 19:12	vote	accept	Alexander Whyte
Sep 14, 2017 at 21:02
Sep 14, 2017 at 6:42	history	edited	Glen_b	CC BY-SA 3.0	added 4 characters in body
Sep 14, 2017 at 6:21	history	answered	Glen_b	CC BY-SA 3.0